Goal: The fifth grade student will be able to recognize and generate equivalent forms of fractions, using fractional forms of the number one. (NCTM Standards of Numbers and Operations from the Principles and Standards for School Mathematics, 2000, Grades 3-5, p. 392, and Academic Content Standards: K-12 Mathematics for Ohio, Number, Number Sense and Operations Standard, Grade Four, Standard #1, 2001, p. 137 and Grade 5, p. 63.) The students will already have learned how to compare fractions for relative size.

This lesson can also be used for remedial work in Integrated Math I (9^th Grade).

Pre-Activity: Fractions of Pizzas

Since many people like to eat pizza, we are going to use drawings of pizzas in this activity.

Let’s compare some simple pizza drawings:

Pizza A is a whole pizza, Pizza B was cut into Pizza C was cut into

not cut into slices. two pieces of the same size; eight identical pieces;

one piece was eaten, and only only four are left.

one piece remains.

1. Compare the amount of pizza that remains of Pizza B as compared to that of Pizza C. ________________________________________________________________________________________________________________________________________________

2. What fraction best describes the amount of Pizza B that remains?_______________

3. What fraction of Pizza C remains?_______________________________________

4. Based on your answer for Problem 1, compare the fractions that you found in Problems 2 and 3. What can you say about the value of these fractions?

(Hint: Are some of the fractions larger than the others?)

________________________________________________________________________________________________________________________________________________

ACTIVITY I: Multiplying Numerator and Denominator of Fractions by the Same Nonzero Number

Let’s look at some fractions in table form:

(Y)		(Y)
(G)			(G)
(O)		(O)

NOTE: The letters in the colored cells are labeled Y for yellow, G for green and O for orange, for people who do not distinguish colors.

Fill out the table above, multiplying the fractions found in the first column by the fractions at the top of the other columns. Notice that you are multiplying both numerator and denominator of the given fractions by the same number.

The first square has been completed as an example: = .

What is the integer value for each of the following fractions? ________

=______; = _______; = _______ ;

What can you conclude about the integer value of all of the fractions in the first row? _______________________________________________________________________

Do you think that multiplying the fractions in the first column by numbers such as and changed the value of the fractions listed in the first column? Explain your answer. ________________________________________________________________________________________________________________________________________________

So, in general, what should happen to the value of a fraction when you multiply by fractions such as, and ?

________________________________________________________________________

Further Investigation:

Look at the yellow squares (Y). The fractions in them should look familiar; they are the same ones that represented the leftover pieces of pizzas B and C.

Based on what you discovered earlier about the fractions of pizzas B and C, what do you conclude about the value of the fractions in the yellow squares? How big are they in relation to each other?

________________________________________________________________________________________________________________________________________________

Look at the fractions in the green squares in your chart (G). The fraction is shown below:

Shade in the number of squares in the rectangle below that you think best represents the fraction you wrote in the second green square:

What can you conclude about the value of the fractions in the green squares?

________________________________________________________________________________________________________________________________________________

Now look at the fractions in the orange squares in your chart (O). The fraction is shown below:

Shade in the number of squares in the rectangle below that you think best represents the fraction you wrote in the second orange square.

What can you conclude about the fractions in the orange squares? ___________________________________________________________________________________________

_____________________________________________________________________

EXTENSION: Look at your chart again. Based on what you have just learned, do you think that is equivalent to? Why or why not? ________________________________________________________________________________________________________________________________________________

If you did not have a chart, how could you find the answer to the questions above?

________________________________________________________________________________________________________________________________________________

Based on what you have just discovered, what can you conclude about the fractions in any single row of your chart?

________________________________________________________________________________________________________________________________________________

Since you multiplied the numerator and denominator of the fractions in the first column of your chart by different nonzero numbers to get the rest of the fractions in that row, summarize what you have discovered about multiplying the numerator and denominator of a fraction by the same nonzero integer, such as 2 or 3:

________________________________________________________________________________________________________________________________________________

How can we find the value of the fraction?

(Hint: Remember that any fraction can also be called a division expression, a divided by b, as long as does not equal zero.) ________________________________________________________________________________________________________________________________________________

Fill out the third column of the chart below:

Nonzero integer	Fraction,	= ?
1
2
3
12

Look at what you wrote down in the third column of the chart, and then answer the following questions:

Compare the answers that you wrote in the third column of the chart. How big are they in relation to each other? Is one value bigger than the others? Explain your answer.

________________________________________________________________________________________________________________________________________________

What can you conclude about the value of, as long as is a nonzero integer?

________________________________________________________________________________________________________________________________________________

Based on what you have discovered in this activity, explain how to obtain equivalent fractions using multiplication:

________________________________________________________________________________________________________________________________________________

Based on what you have just discovered about multiplying fractions by (where n 0), hypothesize what you think will happen when you divide fractions by a form of one:

________________________________________________________________________________________________________________________________________________

ACTIVITY II: Dividing Numerator and Denominator of a Fraction by the Same Nonzero Number

Fill out the table below, dividing the fractions found in the first column by the fractions at the top of the other columns. The first square has been completed as an example:

(R)			(R)
(B)			(B)
(G)			(G)
(Y)			(Y)

NOTE: The letters in the colored cells are labeled R for red, B for blue, G for green, and Y for yellow, for people who do not distinguish colors.

What is the decimal value for each fraction in a red square (R)?

The fraction in the first square, = 6 divided by 12 = _____ (decimal)

The other fraction in a red square is ____ = ________ (decimal)

What can you conclude about the decimal value of all of the fractions in the row with the red squares? _______________________________________________________________________

_________________________________________________________________________________________________________________________________________________________

What is the decimal value for each fraction in a blue square (B)?

= 6 divided by 18 = _____ (decimal)

The other fraction in a blue square is ____ = _______ (decimal)

What can you conclude about the decimal value of all of the fractions in the row with the blue squares? _______________________________________________________________________

What is the decimal value for each fraction in a green square (G)?

= 6 divided by 24 = _____ (decimal)

The other fraction in a green square is ____ = ________ (decimal)

What can you conclude about the decimal value of all of the fractions in the row with the green squares? _______________________________________________________________________

What is the decimal value for each fraction in a yellow square (Y)?

= 12 divided by 18 = _____ (decimal)

The other fraction in a yellow square is ____ = _______ (decimal)

What can you conclude about the decimal value of all of the fractions in the row with the yellow squares? _______________________________________________________________________

Based on what you have just discovered, what can you conclude about the fractions in any single row of this chart?

________________________________________________________________________

Summarize what you have learned about dividing fractions by a form of one, such as, or, in general,, where n is any nonzero integer:

_____________________________________________________________________________________________________________________________________________________________

What do you conclude about multiplying or dividing any number (including fractions) by, where n is any nonzero integer?

____________________________________________________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________________________________________________

APPLICATION:

A. Given the fraction, find four equivalent fractions with numerators larger than three: ______, ______, ______. ______.

B. Given the fraction, find two equivalent fractions with denominators smaller than 16: ______, ______.

EXTENSION:

Fill in Column Two of the table below (under Fraction Two), with your answers from Question A above, and then complete the chart:

Fraction One	Fraction Two	Numerator of Fraction One Times Denominator of Fraction Two	Numerator of Fraction Two Times Denominator of Fraction One

What do you notice about the numbers in columns three and four?

________________________________________________________________________________________________________________________________________________

Does this give you a way to see if two fractions are equivalent? Explain your answer:

________________________________________________________________________________________________________________________________________________

Using the strategy you just discovered, prove or disprove that is equivalent to: ________________________________________________________________________________________________________________________________________________

GENERALIZE:

A. Name two ways to find equivalent fractions:

1. __________________________________________________________________

2. __________________________________________________________________

B. Describe two ways to see if two fractions are equivalent:

1. __________________________________________________________________

2. __________________________________________________________________

EXTENSION:

Why do you think that n must be nonzero, when using n to find equivalent fractions? Let’s explore this idea a little deeper:

What happens when you multiply a number by zero?

________________________________________________________________________________________________________________________________________________

Would multiplying two fractions by zero give you equivalent fractions? Explain your answer:

________________________________________________________________________________________________________________________________________________

Why can’t you divide both fractions by zero?

________________________________________________________________________________________________________________________________________________

Based on your answers above, why must n have to be a nonzero number?

________________________________________________________________________________________________________________________________________________

FURTHER EXTENSION:

Can you think of any practical uses for equivalent fractions? Describe them below:

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